Railroad Math

Introduction

I was listening to an advertisement where CSX made the claim that they move 1 ton of freight 423 miles for 1 gallon of fuel. This is an interesting measure of efficiency. Let's see if we can confirm this using a couple of different analysis approaches.

Top-Down Approach

CSX issues quarterly reports, which contains data that we can use to estimate the miles per gallon-ton of freight. This forum article has some interesting data from the 4Q2007 CSX financial statement.

  • CSX moved 253 billion revenue ton-miles of goods in the 12 months ending 12/31/07.
  • During this period, CSX consumed 569 million gallons of diesel #2 fuel.

This makes computing the efficiency eff of a train (mile-ton per gallon) easy, which is shown in Equation 1.

Eq. 1 \mathit{eff}=\frac{253\text{E9}\cdot \text{mile}\cdot \text{ton}}{569\text{E9}\cdot \text{gal}}=445\frac{\text{mile}\cdot \text{ton}}{\text{gal}}

This number is close to what CSX is using its advertisement. So their number is credible based on their business statment.

Bottom-Up Approach

It is a bit more difficult to look at the problem from the standpoint of friction and energy, but let's take a wack at it.

First, let's gather some data.

  • Energy per gallon of #2 diesel fuel is 138,700 BTU/US gal (Source)
  • Efficiency of a diesel engine is ~46% (Source and Source)
  • Diesel to rail conversion efficiency of 80% (Source)
    There is some loss of power due to transmission inefficiency in the diesel-electrical-rail transfer of power. I am using the value of 80% for the transmission efficiency based on a reference from the 1950s that was comparing steam to diesel-electric locomotives. This number is probably out of date, but is a reasonable start for a rough estimate.
  • Train expends 20 lb of pulling force per ton of load (Source)
    This number is subject to variation due to track condition, weather, grade, and curvature of the track. I am assuming an average value that is in the ballpark, but could easily be off by ±20% or more. Remember, we are just trying to determine if the CSX efficiency number is reasonable.

I would propose that one simple model would be to equate the energy dissipated against the rolling resistance of the train to the energy available from a gallon of diesel fuel.

Eq. 2 {{F}_{FrictionPerTon}}\cdot d={{E}_{FuelOilPerGal}}\cdot {{e}_{Diesel}} \cdot e_{c}

Where d is the distance traveled, FFrictionPerTon is the resistance of a ton of load (= 20 lb per ton), EFuelOilPerGal is the energy per gallon of #2 diesel fuel (=138,700 BTU/US gal), eDiesel is the efficiency of a modern diesel engine (=46%), and ec is the diesel-to-rail conversion efficiency (=80%).

We can solve Equation 2 for d and substitute our assumed values.

Eq. 3 d=\frac{{{E}_{FuelOilPerGal}}\cdot {{e}_{Diesel}} \cdot e_c }{{{F}_{FrictionPerTon}}}=376\frac{\text{mile}\cdot \text{ton}}{\text{gal}}

This value is close enough that feel I have verified the CSX number from the bottom up.

Conclusion

Moving one ton of freight 423 miles on one gallon of fuel seems like a reasonable value. This exercise really shows the efficiency of moving material in bulk.

 
This entry was posted in General Science. Bookmark the permalink.

5 Responses to Railroad Math

  1. irstuff says:

    Great! Thanks for doing this calc. I had a similar desire, but didn't find the friction value.

    I have just one teensy nitnoid. Your units on the last equation show miles*ton/gal, but it really should be mile*ton/gal; plural units do not belong in equations. I'm surprised that that Mathcad accepted that.

     
  2. Joel says:

    Could you be high because there is some loss in converting from the diesel motor to electricity?

     
    • mathscinotes says:

      Probably. I have found several sources that said that locomotive diesels operate about 46%. The Wikipedia reports that the most efficient diesel engine operates at 54%. So I think I know the range of diesel efficiency.

      However, I found little information on the conversion efficiency of diesel power to rail power. One source commented that the conversion from diesel to rail cost about 20% (I had to back that out). That would mean the overall efficiency was about 46%·80%=37%. This would drop my efficiency estimate down to 370 mile·ton/gal. I will update the post to reflect this updated estimate.

      Good catch.

       
  3. Pingback: Fuel Efficiency Math | Math Encounters Blog

Leave a Reply

Your email address will not be published. Required fields are marked *